Chapter 2 Discrete-Time Signals and systems
Chapter 2 Discrete-Time Signals and Systems
82.1 Discrete-Time Signals: Time-Domain Representation Signals represented as sequences of numbers, called samples Sample value of a typi ical signal or sequence denoted as xn with n being an nteger in the range -oos n <oo xn defined only for integer values of n and undefined for noninteger values of n Discrete-time signal represented by xng
§2.1 Discrete-Time Signals: Time-Domain Representation • Signals represented as sequences of numbers, called samples • Sample value of a typical signal or sequence denoted as x[n] with n being an integer in the range -∞≤ n ≤∞ • x[n] defined only for integer values of n and undefined for noninteger values of n • Discrete-time signal represented by {x[n]}
82.1 Discrete-Time Signals: Time-Domain Representation Discrete-time signal may also be written as a sequence of numbers inside braces {xn}={,0.2,2.2,1.1,0.2,3.7,2.9, °· The arrow is placed under the sample at time index n=o In the above, x-1=-0.2, x 0=2.2 X1F1.1, etc
§2.1 Discrete-Time Signals: Time-Domain Representation • Discrete-time signal may also be written as a sequence of numbers inside braces: {x[n]}={…,-0.2,2.2,1.1,0.2,-3.7,2.9,…} ↑ • The arrow is placed under the sample at time index n = 0 • In the above, x[-1]= -0.2, x[0]=2.2, x[1]=1.1, etc
82.1 Discrete-Time Signals: Time-Domain Representation Graphical representation of a discrete-time signal with real valued samples is as shown below. x[-5 x 3456789101112 10-9-8-7-6-5-4-3-2-10120 1314151617
§2.1 Discrete-Time Signals: Time-Domain Representation • Graphical representation of a discrete-time signal with realvalued samples is as shown below:
82.1 Discrete-Time Signals: Time-Domain Representation In some applications, a discrete-time sequence xn may be generated by periodically sampling a continuous-time signalxa (t) at uniform intervals of time x4-57 a 3T -5T-37-70T x2(37)
§2.1 Discrete-Time Signals: Time-Domain Representation In some applications, a discrete-time sequence {x[n]} may be generated by periodically sampling a continuous-time signal xa (t) at uniform intervals of time
82.1 Discrete-Time Signals: Time-Domain Representation Here, n-th sample is given by xn]=x()n=Xa(mI,n=.,2,-1,0, tnT=X The spacing t between two consecutive samples is called the sampling interval or sampling period Reciprocal of sampling interval t denoted as FT, is called the sampling frequency: F=1/T
§2.1 Discrete-Time Signals: Time-Domain Representation • Here, n-th sample is given by x[n]=xa (t) |t=nT=xa (nT), n=…,-2,-1,0,1,… • The spacing T between two consecutive samples is called the sampling interval or sampling period • Reciprocal of sampling interval T, denoted as FT , is called the sampling frequency: FT=1/T
82.1 Discrete-Time Signals: Time-Domain Representation Unit of sampling frequency is cycles per second, or hertz(Hz), if T is in seconds Whether or not the sequence xnb has been obtained by sampling, the quantity x n is called the n-th sample of the sequence xn is a real sequence, if the n-th sample xn is real for all values of n Otherwise, xn is a complex sequence
§2.1 Discrete-Time Signals: Time-Domain Representation • Unit of sampling frequency is cycles per second, or hertz (Hz), if T is in seconds • Whether or not the sequence {x[n]} has been obtained by sampling, the quantity x[n] is called the n-th sample of the sequence • {x[n]} is a real sequence, if the n-th sample x[n] is real for all values of n • Otherwise, {x[n]} is a complex sequence
82.1 Discrete-Time Signals: Time-Domain Representation a complex sequence xn can be written as xn=xenl+jimiN) where and x. are the real and re imaginary parts of xn The complex conjugate sequence of xn is given by x*nxrela-jximIng Often the braces are ignored to denote a sequence if there is no ambiguity
§2.1 Discrete-Time Signals: Time-Domain Representation • A complex sequence {x[n]} can be written as {x[n]}={xre[n]}+j{xim[n]} where xre and xim are the real and imaginary parts of x[n] • The complex conjugate sequence of {x[n]} is given by {x*[n]}={xre[n]} - j{xim [n]} • Often the braces are ignored to denote a sequence if there is no ambiguity
82.1 Discrete-Time Signals: Time-Domain Representation Example-=cos0 25n is a real sequence ynej j0.ni is a complex sequence · We can write n)=cos03n jsin03n) ={c003n}+j{sin0.3n where rela= cos0 3ng imla= sin.3ng
§2.1 Discrete-Time Signals: Time-Domain Representation • Example - {x[n]}={cos0.25n} is a real sequence {y[n]}={ej0.3n} is a complex sequence • We can write {y[n]}={cos0.3n + jsin0.3n} = {cos0.3n} + j{sin0.3n} where {yre[n]}={cos0.3n} {yim[n]}={sin0.3n}
82.1 Discrete-Time Signals: Time-Domain Representation Two types of discrete-time signals: Sampled-data signals in which samples are continuous-valued Digital signals in which samples are discrete-valued Signals in a practical digital signal processing system are digital signals obtained by quantizing the sample values either by rounding or truncation
§2.1 Discrete-Time Signals: Time-Domain Representation • Two types of discrete-time signals: - Sampled-data signals in which samples are continuous-valued - Digital signals in which samples are discrete-valued • Signals in a practical digital signal processing system are digital signals obtained by quantizing the sample values either by rounding or truncation
